AP Physics 1- 7.1 Defining Simple Harmonic Motion (SHM)- Study Notes- New Syllabus
AP Physics 1-7.1 Defining Simple Harmonic Motion (SHM) – Study Notes
AP Physics 1-7.1 Defining Simple Harmonic Motion (SHM) – Study Notes -AP Physics 1 – per latest Syllabus.
Key Concepts:
- Simple Harmonic Motion (SHM)
Simple Harmonic Motion (SHM)
Simple harmonic motion is a special type of periodic motion in which an object oscillates about an equilibrium position in a regular, repeating manner.
Restoring Force in SHM:
SHM occurs when the magnitude of the restoring force acting on an object is directly proportional to the displacement from the equilibrium position and is directed opposite to the displacement. This restoring force always acts to bring the object back toward its equilibrium.
Mathematical Formulation:
Let \( x \) be the displacement from equilibrium and \( k \) be the force constant (spring constant for a spring). The restoring force \( F \) is given by:
\( F = – k x \)
Using Newton’s second law \( F = m a \), the acceleration \( a \) of the object is:
\( m a = – k x \implies a = – \dfrac{k}{m} x \)
Key Concepts:
- Restoring Force: A force that is always directed opposite to displacement and proportional to it.
- Equilibrium Position: The point at which the net force on the object is zero. The object oscillates around this position.
- Pendulum Example: For a pendulum with a small angular displacement, the restoring torque is proportional to the angular displacement (\( \tau \propto -\theta \)), allowing it to be modeled as SHM.
Example :
A spring with force constant \( k = 150 \, \text{N/m} \) is stretched by \( x = 0.2 \, \text{m} \). Find the restoring force acting on the spring.
▶️Answer/Explanation
Restoring force: \( F = – k x = – (150)(0.2) = -30 \, \text{N} \)
The negative sign indicates the force acts opposite to the displacement.
Answer: \( F = 30 \, \text{N} \) (toward equilibrium)
Example :
A mass of \( m = 0.5 \, \text{kg} \) is attached to a spring with \( k = 200 \, \text{N/m} \). If the displacement from equilibrium is \( x = 0.1 \, \text{m} \), find the acceleration of the mass.
▶️Answer/Explanation
Acceleration: \( a = – \dfrac{k}{m} x = – \dfrac{200}{0.5} \cdot 0.1 = -40 \, \text{m/s}^2 \)
The negative sign indicates the acceleration is directed toward the equilibrium position.
Answer: \( a = 40 \, \text{m/s}^2 \) (toward equilibrium)